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Sunday, April 8, 2012

Knots

Knots are entanglement. They serve to bind, to secure, to tie. But the complex bonds that they represent are more than just bonds and complexity. There is actually a science to them.

Their overlapping, interweaving character is what makes them secure. Because of it, they don't fall apart. If you take one strand, you can't pull it away because it's entangled in the knot. This is one simple definition of a knot: no one strand can be removed without untying it.

In mathematics, a knot is a loop. The loop is officially embedded in three-dimensional space, but you can also embed it in the plane, if you consider, in addition, the crossing.

The overhand knot shown above is actually a trefoil knot with one cut. It is crucial to realize that any knot can be untied if the strand is cut.

In the real world, knots may be composed of more than one loop as well. An instance of this is two loops interlocking. Or the small weave is another good example.

Knots are important to sailors because ropes tie all things on a ship. Sometimes, like in the figure eight bend knot which can connect two ends of rope, they are designed to be strong, and yet untied in an instance so the two ends of rope can be disconnected.

In mathematics, again, there is a notion of a prime knot. This is a knot consisting of a loop that is indecomposable into a simpler prime knot. Each prime knot is defined by the number and configuration of their crossings. Cut the loop once and you can untie it.

The trefoil knot is the only prime knot with three crossings. It is one of the prototypes for the Valknut (along with the Borromean rings), a knot from Norse mythology. In the mathematical world, it is the simplest prime knot.

A simple figure-eight may be twisted and turned into a simple loop, called the unknot.

If you cut it at any place, you end up with the standard overhand knot, which is a knot in the middle of a single strand, used to thicken a strand so it might stay in place, for instance. This is called a stopper knot. A better stopper knot is the figure eight knot. Over, under, around, and through.

Some knots are really just intertwining of multiple strands. But these are not prime knots. A typical example of this is the simple 3-braid. This consists of three strands that are interwoven.

Pippi Longstocking (Pippi Långstrump in Sweden) probably braids her red pigtails this way. As did the Viking women more than a thousand years ago on Gotland.

Given three fibers, pretty much everybody knows how to make a braid. But really a braid can be made out of any number of fibers. In fact, the copper wire shielding on coaxial cable is braided in a pattern that wraps around in a circle, so it represents a higher-order of braid with a cylindrical connectivity. This kind of braid is less embeddable in the plane than, say, the 3-braid shown here.

In some ways, braids are a degenerate form of a weaving. But knots aren't really.

Knots are interesting to tie from rope. They have several uses. For instance, you can make a loop at the end of a rope for securing the rope to an object. This is a common knot, the slip knot. It has the distinction for being less secure than other knots made for the same purpose. For instance, the noose (shown) is a more secure loop end.

This is because, when you make a slip knot, it is important to take care that the part that slips is not the loose end. Otherwise it will slip right out. As you can see here, you make the connected end the part that slips and when you pull on it, it will become tighter.

To a point. And then the loose end will eventually pull through. Which is why a real noose additionally secures the loose end in some way. Using a figure eight knot as a stopper knot on the loose end is a good way to make sure it won't slip through eventually.

This is a very simple utilitarian knot.

But if you really need a good stopper knot at the end of a rope, try the double overhand knot. This knot is a bit larger than the overhand knot, more secure than the figure eight knot, and it's also probably easier to untie when you need to.

The double overhand knot is like an overhand knot, but you pass the loose end through one more time than usual. Once you have made the knot, then pull it tight and it makes a tight ball at the end of your rope that is very secure. It looks like the knot shown here. It makes one of the best stopper knots. When you don't want a rope to pull through a hole, you use this one.

It's secure enough to use when climbing, for instance.

If you want to learn how to tie proper knots, you can use the animated knot site, or their animated knot how-to apps.

Cryptography?

Anyway, knots are fun, and they have a mathematical basis that is even useful for cryptography. Imagine a digital signature scheme using a braid group, for instance. You can come up with standard notations for the prime knots also. More complex knots can be found to be decomposed into connected sums of prime knots. Unfortunately this is hard, like factoring numbers. And this is another reason why knot theory is sometimes used for cryptography.

Knots form groups and groups can be the source of endless interest.

Knot theory has often been identified by physicists as a basis for the process of quantum entanglement and other natural processes that occur at the subatomic level.